modeling the asymmetry of stock movements using price ranges

Modeling the Asymmetry of Stock Movements Using Price

Ranges

Ray Y. ChouAcademia Sinica

“ The 2002 NTU International Finance Conference” Taipei. May 24-25, 2002

Motivation

• Provide separate dynamic models for the upward-range and the downward-range to allow for asymmetries.

• Factors driving the upward movements and the downward movements maybe different.

• Upward range applications: market rallies, call options, historical new highs, limit order to sell

• Downward range applications: Value-at-Risk, put options, limit order to buy

Main Results• ACARR is similar to CARR and ACD but

with a different limiting distribution and with new interpretations and implications.

• Properties: QMLE, Distribution• Empirical results using daily S&P500 index

show asymmetry in dynamics, leverage effect, periodic patterns and interactions of upward and downward movements.

• Volatility forecast accuracy: ACARR>CARR>GARCH

Range as a measure of the “realized volatility”

• Simpler and more natural than the sum-squared-returns (measuring the integrated volatility) of Anderson et.al.(2000)

• Parkinson (1980) and others have established the efficiency gain of range over standard method in estimating volatilities

• Chou (2001) proposed CARR, a dynamic model for range with satisfactory performance

Discrete sampling from a continuous process

Let Pt be the logarithmic price of a speculative asset observed at

time t, t =1,2,…T. Pt are taken to be realizations of a price

process {P}, which is assumed to be a continuous process.

Let OPENtP , CLOSE

tP , HIGHtP , LOW

tP be the opening, closing, high and

low prices, in natural logarithm, between t-1 and t.

Upward range and downward range

Define UPRt and DWNRt as the differences between

the daily highs, daily lows and the opening price

respectively, at time t, i.e.,

OPENt

HIGHtt PPUPR

).( OPENt

LOWtt PPDWNR

Range and one-sided ranges

Note that these two one-sided ranges, UPR t and

DWNR t, represent the maximum returns and the

min imum returns (in percentage size) respectively

over the unit time interval [t-1, t]. Further, the

range R t, is defined to be

LO W

tH IG H

tt PPR , or

ttt DWNRUPRR .

The Conditional Autoregressive Range Expectation (CARR) model in Chou

(2001)

tttR

jt

q

jjit

p

iit R

11

t ~ iid f(.)

The Asymmetric Conditional Autoregressive Range Expectation -ACARR(p,q) model

ut

uttUPR

dt

dttDWNR

u

jt

q

j

ujit

p

i

ui

uut UPR

11

d

jt

q

j

djit

p

i

di

ddt DWNR

11

Distribution assumptions in ACARR

ut ~ iid fu(.), d

t ~ iid fd(.)

It is shown that the limiting distributions fu(.) and fd(.) are identical but are different from the error distribution for the CARE model in Chou (2000b).

ACARRX(p,q) – ACARR(p,q) with exogenous variables

Let be some variables measurable at time t-1, the ACAREX(p,q) model is defined to be

,1111

ltl

L

l

u

jt

q

j

ujit

p

i

ui

uut XUPR

,1111

ltl

L

l

d

jt

q

j

djit

p

i

di

ddt XDWNR

Explanatory variables in the ACARRX(p,q) model

• Lagged returns – leverage effect• Periodic (weekday) pattern• Transaction volumes• Interaction tems – lagged DWNR in expecte

d UPR and lagged UPR in expected DWNR

Properties of ACARR

• Same as ACD of Engle and Russell (1998) but with a known limiting distribution for the error term

• A conditional mean model • An asymmetric model for volatilities

Sources of asymmetry for an ACARRX(1,1) model

– short term shock impact – long term persistence of shocks – speed of mean-reverting ‘s – effects of leverage, periodic pattern,

interaction terms, among others

A special case of ACARR: Exponential ACARR(1,1) or EACARR(1,1)

• It’s useful to consider the exponential case for f(.), the distribution of the normalized range or the disturbance.

• Like GARCH models, a simple (p=1, q=1) specification works for many empirical examples.

ACARR vs. ACD identical formula

• ACARR• Range data, positive

valued, with fixed sample interval

• QMLE with EACARR• Known limiting

distribution• A new volatility

model

• ACD• Duration data, positive

valued, with non-fixed sample interval

• QMLE with EACD• Unknown limiting

distribution• Hazard rate

interpretation

The QMLE property

• Assuming any general density function f(.) for the disturbance term t, the parameters in ACARR can be estimated consistently by estimating an exponential-ACARR model.

• Proof: see Engle and Russell (1998), p.1135

The QMLE Estimation

• Consistent standard errors are obtained by employing the robust covariance method in Bollerslev and Wooldridge (1987).

• See Engle and Russell (1998).

Empirical example: S&P500 daily index

• Sample period: 1962/01/03 – 2000/08/25• Data source: Yahoo.com• Models used: EACARR(1,1), EACARRX(p,q)• Both daily and weekly observations are used

for estimation• Forecast comparison of CARR and ACARR

-30

-20

-10

0

10

2000 4000 6000 8000

UPR DWNR

Figure 1: Daily UPR and Daily DWNR, S&P500, 1962/1-2000/8

Series: MAXSample 1 9700Observations 9700

Mean 0.737026Median 0.635628Maximum 9.052741Minimum 0.000000Std. Dev. 0.621222Skewness 2.110659Kurtosis 15.76462

Jarque-Bera 73055.13Probability 0.000000

0

400

800

1200

1600

2000

2400

0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75

Series: MAXSample 1 9700Observations 9700



Figure 2: Daily UPR of S&P500 Index, 1962/1-2000/8

Series: AMINSample 1 9700Observations 9700


Jarque-Bera 6376152.Probability 0.000000

0

1000

2000

3000

4000

5000

6000

7000

8000

0 2 4 6 8 10 12 14 16 18 20 22




Figure 3: Daily DWNR of S&P500 Index, Unsigned, 1962/1-2000/8




0

400

800

1200

1600

2000

2400

0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75




Figure 4: Daily DWNR w/o Crash, Unsigned, 1962/1-2000/8

Nobs Mean Median Max Min Std Dev 1 12 Q(12)

Full sampleRANGE 1/2/62-8/25/00 9700 1.464 1.407 22.904 0.145 0.76 0.629 0.575 0.443 30874UPR 1/2/62-8/25/00 9700 0.737 0.636 9.053 0 0.621 0.308 0.147 0.172 3631DWNR 1/2/62-8/25/00 9700 -0.727 -0.598 0 -22.9 0.681 0.326 0.181 0.162 4320

Before structural shiftRANGE 1/2/62-4/20/82 5061 1.753 1.643 9.326 0.53 0.565 0.723 0.654 0.554 21802UPR 1/2/62-4/19/82 5061 0.889 0.798 8.631 0 0.581 0.335 0.087 0.106 1199DWNR 1/2/62-4/19/82 5061 -0.864 -0.748 0 -6.514 0.559 0.378 0.136 0.163 2427

After structural shiftRANGE 4/21/82-8/25/00 4639 1.15 0.962 22.904 0.146 0.818 0.476 0.414 0.229 11874UPR 4/21/82-8/25/00 4639 0.572 0.404 9.053 0 0.622 0.189 0.089 0.125 651DWNR 4/21/82-8/25/00 4639 -0.578 -0.388 0 -22.9 0.767 0.247 0.147 0.101 994

Table 1: Summary Statistics of the Daily Range, Upward Range and Downward Range of S&P500 Index, 1/2/1962-8/25/2000

0.05

0.10

0.15

0.20

0.25

0.30

0.35

20 40 60 80 100 120 140 160 180 200

RHO_UPR RHO_DWNR RHO_DWNR(w/o crash)

Figure 5: Correlograms of Daily UPR and DWNR

T ab le 2 : Q M L E E st im a tion of AC A RR Using D a ily Upw ar d R an ge of S& P 5 00 Ind ex 1/2 /1 96 2-8 /25 /2 00 0

t ~ iid f( .) E s tima tio n i s c a rr ied ou t us in g th e QM L E m e th od h e nc e it 's equ iva lent to e stim a t in g a n E xp on en tia l AC AR R( X ) (p, q) or a nd E AC AR R (X ) (p ,q) m o de l . N u mbe rs in p a re nthe se s a re t-ra tios ( p-v a lu e s) w ith ro bu st sta n da rd e rr ors fo r the m od el c o ef fi cie nts ( Q sta tist ic s). L L F is th e log l ike l ih oo d fun c tion . A C AR R( 1,1 ) AC A R R( 2,1 ) A CA RR X ( 2,1 )-a A CA RR X ( 2,1 )-b A CA RR X ( 2,1 )-c

L L F -1 20 35 .2 0 -1 20 11 .8 6 -1 19 55 .7 8 -1 19 49 .6 4 -1 19 50 .3 2

c on sta n t 0. 00 2 [ 3.2 16 ] 0. 00 1 [ 3.1 45 ] -0. 00 2 [-0.6 10 ] -0. 00 3 [-0.5 51 ] -0. 00 4 [-0.9 73 ]

UP R( t-1 ) 0 .0 3 [ 8.8 73 ] 0. 14 5 [1 0.8 37 ] 0. 20 3 [1 4.0 30 ] 0. 17 9 [1 1.8 56 ] 0. 18 6 [1 2.8 45 ]

UP R( t-2 ) -0. 12 6 [-9.1 98 ] -0. 11 7 [-9 ..4 48 ] -0. 11 2 [-8.8 79 ] -0. 11 5 [-9.2 45 ]

( t-1 ) 0. 96 8 [ 26 7.9 93 ] 0. 97 8 [ 34 1.9 23 ] 0. 90 3 [ 69.6 43 ] 0. 87 1 [4 8.4 26 ] 0. 87 7 [5 2.9 42 ]

r( t-1 ) -0. 05 7 [-8.4 31 ] -0. 01 8 [-1.9 59 ] -0. 02 3 [-2.7 34 ]

T U E 0. 05 8 [ 3.4 23 ] 0. 05 9 [ 3.4 75 ] 0. 05 9 [ 3.4 81 ]

W E D 0 .0 2 [ 1.2 71 ]

SD 0 .0 00 0 [ 0.2 01 ] -0. 00 3 [-1 .6 47 ]

DW NR (-1 ) 0. 04 6 [ 4.8 68 ] 0. 04 2 [ 4.8 03 ]

Q (12 ) 18 4. 4 [ 0.0 00 ] 2 2. 34 6 [ 0.0 34 ] 2 2. 30 4 [ 0.0 34 ] 2 0. 28 2 [ 0.0 62 ] 2 0. 50 3 [ 0.0 53 ]

tttU P R

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ltl

L

l

u

jt

q

j

ujit

p

i

ui

uut XU P R

T able 3 : Q M L E Es tim at io n o f ACA RR U sin g D aily Do wnw ard Ra ng e of S& P5 00 In dex 1/2 /1 9 62-8 /25 /2 0 00

t ~ iid f(.) Estimatio n is carried ou t usin g th e QM LE m ethod h en ce it's equ ivalent to es tim atin g an Expo n en tial AC A RR (X )(p,q) or and EA CA R R(X)(p ,q) mo del. N um bers in p arentheses ar e t-ratio s(p-values) with rob ust standard er ro rs for the m od el co eff icients ( Q statistics). LLF is the log lik elih oo d fun ctio n. A C AR R(1,1) A C AR R(2,1) AC A RR X(2,1)-a AC A RR X(2,1)-b AC A RR X(2,1)-c

LLF -11 929.39 -11 889.61 -11 87 3.54 -11 86 8.55 -11 870.14

con stan t 0 .0 14 [5.905] 0 .0 04 [4.088] 0 .017 [4.37 3] 0 .0 17 [3.23 5] 0 .0 17 [4.417]

DW N R (t-1) 0 .0 84 [11.83 4] 0 .2 29 [1 6.277] 0 .252 [1 6.123] 0 .2 33 [1 4.77 0] 0 .2 39 [1 6.36 4]

DW N R (t-2) -0 .1 95 [-1 3.48 9] -0 .189 [-1 2.811] -0 .1 85 [-1 2.59 4] -0 .1 86 [-12.897]

t-1 0 .8 97 [1 01 .0 2] 0 .9 61 [1 99 .02] 0 .927 [8 7.639] 0 .9 06 [6 1.21 2] 0 .9 11 [6 3.89 3]

r (t-1) 0 .023 [4.72 1] -0 .0 09 [-1.187]

TU E -0 .0 08 [0.50 3]

W ED -0 .051 [-3.58 2] -0 .0 53 [-3.617] -0 .0 52 [-3.58 7]

SD -0 .002 [-2.12 4] 0 .0 01 [0.90 4]

UP R(-1) 0 .0 37 [4.16 4] 0 .0 28 [5.084]

Q (1 2) 192 .8 [0.000] 1 8.94 [0.009] 22 .2 27 [0.03 5] 14 .4 22 [0.27 5] 14 .7 74 [0.254]

tttD W N R ltl

L

l

d

jt

q

j

djit

p

i

di

ddt XD W N R 1

111

-10

-8

-6

-4

-2

0

2

4

2000 4000 6000 8000

UPR_NEG LAMBDA_UPR

Figure 6: Expected and Observ ed Daily UPR, 1962/1-2000/8

-25

-20

-15

-10

-5

0

5

10

2000 4000 6000 8000

DWNR LAMBDA_DWNR

Figure 7: Expected and Observ ed Daily DWNR, 1962/1-2000/8

Series: ET_MAXSample 1 9700Observations 9700



0

1000

2000

3000

4000

5000

0 5 10 15 20 25 30

Series: ET_MAXSample 1 9700Observations 9700



Figure 8: Histogram of Daily et_UPR, 1962/1-2000/8

Series: ET_MINSample 1 9700Observations 9700



0

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8 10 12

Series: ET_MINSample 1 9700Observations 9700



Figure 9: Histogram of daily et_DWNR, 1962/1-2000/8

0

2

4

6

8

10

0 2 4 6 8 10

ET_UPR

Expo

nent

ial Q

uant

ile

Figure 10: Q-Q Plot of et_UPR

0

2

4

6

8

1 0

0 2 4 6 8 1 0

E T _ D W N R

Exp

onen

tial

Qua

ntil

eF ig u re 1 1 : Q -Q p lo t o f e t-D W N R

Figure 11: Q-Q plot of et-DWNR

Ta b le 4: A C A R R v e rsu s C A R R

In -s am p le V ola ti l ity For ec a st Co m pa r iso n U sin g T hre e M ea su re d V ola ti l it ie s as Be nc h m a rks. T he thre e m e a sur e s of vo la t il i ty a re R N G , R ET S Q a nd A R E T : re spe c tiv ely,

da i ly ra n ge s, sq ua re d -da i ly -re turn s, a n d a bso ul te da i ly- r et u rn. A CA R R (1 ,1) m od e l i s f it t ed fo r th e ra ng e ser ie s a n d a A CA RR m o de l s a re fi t te d fo r the u pw ar d r an ge

an d th e do w nw a rd ra ng e ser ie s. F V (C A R R) (F V (A CA R R )) is th e for e ca st e d v ola ti li t y u s ing CA RR ( A CA RR ). ( F V ( A CA RR )) i s th e fo re ca ste d ra ng e us ing t he su m o f the f orc a st ed u pw a rd r an ge a nd d ow n w a rd ra n ge .

P ro pe r tr an sfo rm a t io ns a r e m ad e fo r a d ju sting the d if fe re nc e b etw e e n a va ria nc e e s tim a to r an d a s ta n da rd -de v ia t io n e st im ator . N u m be rs in p ar en the se s a re t-ra tios .

M V t = a + b F V t(C A R R) + ut M V t = a + c F V t(A C A R R) + ut M V t = a + b F V t(C A R R) + c F V t(A C A R R) + ut

M e as ure d V o la t il i ty E x plan a to ry V a ria ble s c o nsta nt FV (C A R R) FV ( A C A R R) A d j . R -sq. S .E .

R N G -0. 06 7 [-0 .36 6] 1 .0 05 [9 6.2 9] 0 .4 89 0 .5 43

R N G -0 .0 06 [-4 .14 8] 1 .0 47 [ 10 1.0 2] 0 .5 13 0 .5 31

R N G -0 .0 67 [0 .63 2] 0 .0 21 [ 0.4 6] 1 .0 26 [2 1.8 3] 0 .5 13 0 .5 31

RE T SQ -1 .2 03 [-1.3 5] 0 .3 97 [1 4.2 5] 0. 02 5 .7 25

RE T SQ -0 .2 65 [-2.9 4] 0 .4 59 [1 6.0 2] 0 .0 26 5 .7 09

RE T SQ -0 .2 49 [-2.7 6] -0 .1 91 [-2.3 2] 0 .6 44 [ 7.6 1] 0 .0 26 5 .7 08

A R E T 1 .1 42 [ 7.4 1] 0 .3 34 [2 7.0 7] 0. 07 0 .6 42

A R E T 0 .1 13 [ 5.8 5] 0 .3 54 [2 8.2 8] 0 .0 76 0 .6 39

A R E T 0 .1 15 [ 5.9 5] -0 .1 06 [-1.9 1] 0 .4 58 [ 8.0 9] 0 .0 76 0 .6 39

Extensions

• Robust ACARR – Interquartile range• Multivariate ACARR• Nonparametric or semiparametric ACARR• Other data sets and simulations• Long memory ACARR’s – IACARR, FIAC

ARR,…• ACARR and option price models

Conclusion

• ACARR is effective in modeling upward and downward market movements.

• Asymmetry found: dynamics, leverage effect, periodic patterns, interaction terms

• CARR provides more accurate volatility forecasts than GARCH (Chou (2001)) and ACARR gives further improvements.

modeling the asymmetry of stock movements using price ranges

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